Method of Markovian summation for study the repeated flow in queueing tandem M|GI|∞ → GI|∞

The paper presents a mathematical model of queueing tandem M|GI|∞ → GI|∞ with feedback. The service times at the first stage are independent and identically distributed (i.i.d.) with an arbitrary distribution function B1(x). Service times at the second stage are i.i.d. with an arbitrary distribution function B2(x). The problem is to determine the probability distribution of the number of repeated customers (r-flow) during fixed time period. To solve this problem, the Markov summation method was used, which is based on the consideration of Markov processes and the solution of the Kolmogorov equation. In the course of the solution, the so-called local r-flow was studied — the number of r-flow calls generated by one incoming customer received by the system. As a result, an expression is obtained for the characteristic probability distribution function of the number of calls in the local r-flow, which can be used to study queuing systems with a similar service discipline and non-Markov incoming flows. As a result of the study, an expression is obtained for the characteristic probability distribution function of the number of repeated calls to the system at a given time interval during non-stationary regime, which allows one to obtain the probability distribution of the number of calls in the flow under study, as well as its main probability characteristics.

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Branching Process Model applied to the 1/f Noise; Formalization

10.31219/osf.io/2f6r4 ◽

2021 ◽

Author(s):

Takayuki Kobayashi

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Closed Form ◽

Probability Distribution Function ◽

Process Model ◽

Branching Process ◽

Fixed Time ◽

Time Interval ◽

Random Particle ◽

Number Of Particles

In order to discuss the 1⁄f problem, the statistics of branching process of particle in a multiplicative medium are developed by taking account of random particle immigration. The probability distribution function of the number of particles founded in the medium at any fixed time and/or of particles detected in a time interval are obtained in closed form. These results are applied to the case that exactly two particles are produced by branching.

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Position-dependent matter density probability distribution function

Physical Review D ◽

10.1103/physrevd.102.123546 ◽

2020 ◽

Vol 102 (12) ◽

Author(s):

Drew Jamieson ◽

Marilena Loverde

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Probability Distribution Function ◽

Matter Density

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A new distance measure for probability distribution function of mixture type

2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100) ◽

10.1109/icassp.2000.862057 ◽

2002 ◽

Cited By ~ 8

Author(s):

Z. Liu ◽

Q. Huang

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Probability Distribution Function ◽

Distance Measure

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An Online Application for ΔR Calculation

Radiocarbon ◽

10.1017/rdc.2016.117 ◽

2016 ◽

Vol 59 (5) ◽

pp. 1623-1627 ◽

Cited By ~ 25

Author(s):

Ron W Reimer ◽

Paula J Reimer

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Calibration Curve ◽

Probability Distribution Function ◽

Museum Collections ◽

Online Program ◽

Online Application ◽

Radiocarbon Calibration

AbstractA regional offset (ΔR) from the marine radiocarbon calibration curve is widely used in calibration software (e.g. CALIB, OxCal) but often is not calculated correctly. While relatively straightforward for known-age samples, such as mollusks from museum collections or annually banded corals, it is more difficult to calculate ΔR and the uncertainty in ΔR for 14C dates on paired marine and terrestrial samples. Previous researchers have often utilized classical intercept methods that do not account for the full calibrated probability distribution function (pdf). Recently, Soulet (2015) provided R code for calculating reservoir ages using the pdfs, but did not address ΔR and the uncertainty in ΔR. We have developed an online application for performing these calculations for known-age, paired marine and terrestrial 14C dates and U-Th dated corals. This article briefly discusses methods that have been used for calculating ΔR and the uncertainty and describes the online program deltar, which is available free of charge.

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Optimal Taylor–Couette turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.236 ◽

2012 ◽

Vol 706 ◽

pp. 118-149 ◽

Cited By ~ 55

Author(s):

Dennis P. M. van Gils ◽

Sander G. Huisman ◽

Siegfried Grossmann ◽

Chao Sun ◽

Detlef Lohse

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Angular Velocity ◽

Probability Distribution Function ◽

Neutral Line ◽

Velocity Ratio ◽

Outer Cylinder ◽

Taylor Number ◽

Counter Rotation ◽

Taylor Couette

AbstractStrongly turbulent Taylor–Couette flow with independently rotating inner and outer cylinders with a radius ratio of $\eta = 0. 716$ is experimentally studied. From global torque measurements, we analyse the dimensionless angular velocity flux ${\mathit{Nu}}_{\omega } (\mathit{Ta}, a)$ as a function of the Taylor number $\mathit{Ta}$ and the angular velocity ratio $a= \ensuremath{-} {\omega }_{o} / {\omega }_{i} $ in the large-Taylor-number regime $1{0}^{11} \lesssim \mathit{Ta}\lesssim 1{0}^{13} $ and well off the inviscid stability borders (Rayleigh lines) $a= \ensuremath{-} {\eta }^{2} $ for co-rotation and $a= \infty $ for counter-rotation. We analyse the data with the common power-law ansatz for the dimensionless angular velocity transport flux ${\mathit{Nu}}_{\omega } (\mathit{Ta}, a)= f(a)\hspace{0.167em} {\mathit{Ta}}^{\gamma } $, with an amplitude $f(a)$ and an exponent $\gamma $. The data are consistent with one effective exponent $\gamma = 0. 39\pm 0. 03$ for all $a$, but we discuss a possible $a$ dependence in the co- and weakly counter-rotating regimes. The amplitude of the angular velocity flux $f(a)\equiv {\mathit{Nu}}_{\omega } (\mathit{Ta}, a)/ {\mathit{Ta}}^{0. 39} $ is measured to be maximal at slight counter-rotation, namely at an angular velocity ratio of ${a}_{\mathit{opt}} = 0. 33\pm 0. 04$, i.e. along the line ${\omega }_{o} = \ensuremath{-} 0. 33{\omega }_{i} $. This value is theoretically interpreted as the result of a competition between the destabilizing inner cylinder rotation and the stabilizing but shear-enhancing outer cylinder counter-rotation. With the help of laser Doppler anemometry, we provide angular velocity profiles and in particular identify the radial position ${r}_{n} $ of the neutral line, defined by $ \mathop{ \langle \omega ({r}_{n} )\rangle } \nolimits _{t} = 0$ for fixed height $z$. For these large $\mathit{Ta}$ values, the ratio $a\approx 0. 40$, which is close to ${a}_{\mathit{opt}} = 0. 33$, is distinguished by a zero angular velocity gradient $\partial \omega / \partial r= 0$ in the bulk. While for moderate counter-rotation $\ensuremath{-} 0. 40{\omega }_{i} \lesssim {\omega }_{o} \lt 0$, the neutral line still remains close to the outer cylinder and the probability distribution function of the bulk angular velocity is observed to be monomodal. For stronger counter-rotation the neutral line is pushed inwards towards the inner cylinder; in this regime the probability distribution function of the bulk angular velocity becomes bimodal, reflecting intermittent bursts of turbulent structures beyond the neutral line into the outer flow domain, which otherwise is stabilized by the counter-rotating outer cylinder. Finally, a hypothesis is offered allowing a unifying view and consistent interpretation for all these various results.

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